There is more than one way to answer this question. Historically, an Anglican was a member of the Ecclesia Anglicana, or English Church; but presently, an Anglican is a member of the orthodox, global Anglican Communion, which is part of the “one, holy, catholic and apostolic church.” We are Christians rooted in Scripture, theology, and history.
Christianity spread slowly through the British Isles during the first few centuries; and an evangelistic mission to Britain was commissioned in the 6th century by Pope Gregory the Great, with a Benedictine monk named Augustine leading the charge. In 597 AD, Augustine was named the very first Archbishop of Canterbury in present-day England. In time, the English Church consisted of several different strands of Christianity (e.g, British, Roman, and Celtic), resulting in diverse practices in the church, as well as tension in how to relate to the Roman Church.
Though best known for his six marriages, Henry VIII had an important role in terms of establishing the Church of England as a independent institution. He declared that the earthly and sovereign leadership of the Church of England resided in the British monarch, not Rome. His motivation for the split is infamous of course, but he was nevertheless the catalyst for the Reformation in England during the peak of theological reform on the continent.
The Archbishop of Canterbury, Thomas Cranmer, was the leader of the Church of England for most of the English Reformation. He created the first versions of the Book of Common Prayer as a replacement for the Roman Missal. Upon King Edward's succession to the throne, Cranmer pressed for even more theological reform in the church, which ultimately resulted in our Anglican confession, the Thirty-Nine Articles of Religion.
Christ instituted two sacraments, and orthodox Anglicans continue to administer them today to God's people. The Sacrament of the Eucharist (or Lord's Supper) is conducted at every Sunday worship service and is open to all baptized believers. The Sacrament of Baptism is the rite of Christian initiation into the covenant body of believers. And as visible words of the Gospel, both sacraments are signs and seals of God’s grace to us in Christ.
Anglicanism embraces a worship style that has existed for many centuries. Our particular liturgy is from the Book of Common Prayer (BCP), where we utilize a worship service that is saturated with Scripture throughout, climaxing with the sermon and Eucharist. The Anglican liturgy provides us with many benefits, one of which is global unity. When we use the BCP, we are - with one voice - worshipping God with our Anglican brethren around the world.
Anglican polity can be traced back to the early church. Anglicans have bishops, who serve as “overseers” of a group of churches in a region or by affiliation. These bishops elect and serve with an archbishop, who represents a particular province (ours being the ACNA) to the global Anglican Communion. Under the bishops on the local level, priests (also known as teaching elders) and deacons serve and lead God's people.